One-Loop Effective Potential in 2-D Dilaton Gravity on Hyperbolic Plane

The one-loop effective potential in 2-D dilaton gravity in conformal gauge on the topologically non-trivial plane ℝ × S1 and on the hyperbolic plane H2/Γ is calculated. For arbitrary choice of the tree scalar potential it is shown that the one-loop effective potential explicitly depends on the reference metric (through the dependence on the radius of the torus or the radius of H2/Γ). This phenomenon is absent only for some special choice of the tree scalar potential corresponding to the Liouville potential and leading to one-loop uv finite theory. The effective equations are also discussed.